Coherent sources

Coherent sources are light sources that emit waves with a constant phase relationship and identical frequencies. This consistency in phase and frequency ensures that the waves maintain a stable phase difference over time, which is crucial for generating clear and predictable interference patterns. Grasping the concept of coherent sources is essential in optics, wave physics, and numerous technological applications. Their capability to create stable and predictable interference patterns renders them invaluable for both experimental and practical purposes.

1.0Definition of Coherent Sources

Two light sources are considered coherent if they emit light waves of the same frequency and maintain a constant phase difference. Typically, such coherence is achieved when the light sources are derived from a single, common source.

2.0Diagram of Coherent Sources

3.0Methods of Obtaining Coherent Sources

1. Division of Wavefront-In this method, the wavefront is divided into two or more parts by reflection or refraction using mirror, lenses, prism or slits.
2. Division of Amplitude-The amplitude of the incoming beam is divided in two parts either by parallel or refraction. These divided parts reunite after traversing different paths and produce interference.

4.0Conditions for Obtaining Coherent Sources

• To achieve coherence, the two light sources must be derived from a single original source using a method that ensures their relative phase difference remains constant over time.
• The two sources must emit monochromatic light; otherwise, differing colors will create distinct interference patterns and overlapping fringes of varying colors.
• The path difference between the waves reaching the screen from the two sources should be kept small, ideally not exceeding 30 cm. If the path difference is too large, the phase difference will vary, resulting in a loss of interference effects and leading to a general illumination on the screen.

5.0Necessity of Coherent Sources for the production of Interference Pattern

• When two monochromatic waves of intensities $I_1$ and $I_2$ and phase difference $\varphi$ meet at a point, the resultant intensity is given by, $I=I_1+I_2+2\sqrt{I_1{I}_2}\mathrm{Cos}\varphi$

$\Rightarrow 2\sqrt{I_1{I}_2}\mathrm{Cos}\varphi$ is called Interference Factor

• If $\mathrm{Cos}\varphi$ remains constant with time,total intensity at any point will be constant.
• For Intensity Maximum $\mathrm{Cos}\varphi =+1$, For Intensity Minimum $\mathrm{Cos}\varphi =-1$, In this case sources are coherent.
• If the Cos varies over time, taking both positive and negative values, the average value of the cosine will be zero. Consequently, the intensity will be uniform across all points, resulting in general illumination on the observation screen. In this scenario, the two sources are incoherent.

6.0Methods For Producing Coherent Sources

1. Young Double Slit Experiment-The sources $S_1$ and $S_2$ are derived from a common source $\text{ S. }$ Any phase change that occurs in $S_1$ also occurs in $S_2$, ensuring that the relative phase difference between $S_1$ and $S_2$ remains constant over time. Therefore, $S_1$ and $S_2$ function as coherent sources.
2. Fresnel’s Biprism Method-Two coherent sources are created from a single coherent source through refraction.
3. Lloyd’s Mirror Method- A source and its reflected image function as two coherent sources.

7.0Types of Coherence

1. Temporal Coherence

• Temporal coherence describes the correlation of a wave's phase at different times. It gauges how consistently the phase of a wave is maintained over time. High temporal coherence indicates that the wave preserves a stable phase relationship for an extended duration.
• Temporal coherence is closely related to the wavelength of light. A light source with a thin spectral width, such as a single wavelength or a very limited range of wavelengths, exhibits high temporal coherence because its phase remains stable over time.
• A light source with high temporal coherence will exhibit a greater coherence length.
• To achieve stable and clear interference patterns over time, the light sources must possess high temporal coherence. Low temporal coherence leads to rapidly fluctuating phase differences, which results in blurred or unstable interference fringes.

2. Spatial Coherence

• Spatial coherence describes the consistency of the phase of a light wave across different points in space. It indicates how uniform the phase remains over various regions of a wavefront. When spatial coherence is high, the phase of the wave is stable over a broad area, resulting in clear and stable interference patterns.
• Spatial coherence is often affected by the size of the light source. A smaller or more collimated light source generally exhibits higher spatial coherence, as the emitted waves are more uniform across the wavefront.
• A light beam with high spatial coherence can be well-collimated and retain a uniform phase across its cross-sectional area.
• A larger coherence area signifies higher spatial coherence. The coherence area is inversely proportional to the size of the light source and can be measured by the spatial coherence length.

Note-High spatial coherence enables the creation of sharp and distinct interference fringes and patterns. Conversely, low spatial coherence causes fringes to become blurred or poorly defined due to variations in phase difference across the wavefront.

8.0Solved Questions on Coherent Sources

Q-1.Why is no interference pattern observed when two coherent sources are

1. Infinitely close to each other
2. Far apart from each other

Solution: $\beta =\frac{\lambda D}{d}\Rightarrow \beta \propto \frac{1}{d}$

1. When two coherent sources are positioned extremely close together, the fringe width becomes very large. In such cases, a single fringe might cover the entire screen, making the interference pattern difficult to observe.
2. As the distance between the sources increases, the fringe width decreases. At very large distances, the fringe width becomes so small that it is difficult to detect, making the interference pattern challenging to observe.

Q-2. Two identical but independent monochromatic sources of light cannot be coherent?

Solution: Light wave emitted by an ordinary source say Sodium Lamp undergoes abrupt phase changes of order $10^{-8}s$, Therefore, the light waves emitted by two separate light sources will generally lack a consistent phase relationship, resulting in incoherence.